Download - Section 2.3 Linear Functions and Slopes
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Section 2.3Linear Functions and Slopes
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The Slope of a Line
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change in y 5 1 6 6
change in x 2 3 5 5m or
Find the slope of the line that passes
through (-2,5) and (3,-1)
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Example
Find the slope of the line passing through the pair of points. (5,-2) and (-1,7)
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The Point-Slope Form
of the Equation of a Line
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1 1
Write the point-slope form of the equation of the
line with slope of 3 that passes through (-1,2).
Substitute into the point-slope form; y-y ( )
2 3( 1)
2 3( 1)
m x x
y x
y x
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Solving in both forms
• A.Write the equation in point slope form of the line with slope 4 that passes through the point (4,-3). B.Then solve the equation for y
y1x1
y-(-3) = 4(x-4) Substituting the values into the euation
y+3 = 4(x-4)This is Point Slope Form. Apply the distributive property for the parentheses. This will give us the slope intercept form. (The equation is solved for y.)
-3 -3
y= 4(x-4)-3-3y= 4x-16-3
• y-y1 = m(x-x1)
(slope intercept form)
Y=4x-19
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Example
Write the point slope form of the equation of the line with slope of -4 that passes through (2,5). Then solve for y.
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If you are given two points
and you need to write an
equation in point-slope
form, then you can use
either point for (x1,y1).
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2 1
2 1
Write the point-slope form of the equation of the
line that passes through (-1,2) and (-4,5). Then
solve for y.
5 2 3First: Find the slope. 1
4 1 3
Second: Substitue into the point-slope fo
y y
x x
1 1
rm.
y-y ( )
5 1( 4)
Third: Solve for y.
5 1( 4)
5 1 4
y=-1x+1
m x x
y x
y x
y x
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Example
Write the point slope form of the equation of the line that passes through (2,5) and (-1,0). Then solve for y.
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The Slope-Intercept Form
of the Equation of a Line
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Point Slope FormFor a nonvertical line with slope m that passes through (x1,y1) the equation is
y-y1 = m(x-x1)
Example: slope = -3
point on the line(-1,-2)
Y-(-2)= -3(x-(-1))
Y+2= -3(x+1)
Slope Intercept FormFor a nonvertical line with slope m and y-intercept b the equation is y=mx+b
Example: slope =2
y-intercept of 6
Y=2x +6
Two forms for Equations of Lines
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Graph the linear equation y= 2/3x+4
x
y
First: Plot the y-intercept of 4Rise by 2 unitsRun ( go to the right) by 3 units.Plot the second point (3, 6)Connect the two points with a straight edge or ruler.
(0,4)
(3,6)
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Example
Graph the linear equation y= -3x+5
x
y
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Example
x
y
1Graph the linear equation y= 3
2x
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Equations of Horizontal
and Vertical Lines
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Example Graph x=4. Graph y=-2
x
y
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The General Form
of the Equation of a Line
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Find the slope and the y intercept of the line.
4 5 20 0
4 5 20
-5y=-4x-20
-5y 4 20
-5 5 54
y= 45
x y
x y
x
x
Y intercept
slope
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Example
Find the slope and the y intercept of the line whose equation is 2x+5y-10=0.
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Using Intercepts to Graph
Ax + By + C = 0
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Find x and y intercepts to graph a line 6x-2y=12
X intercept so let y=0 Y intercept so let x=000
6x-2(0)=12
6x=12
X=2(2,0)
Y=-6(0,-6)
-2y=12
x
y
6(0)-2y=12
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X intercept - Let y=0
4x-3 0 6 0
4x-6=0
4x=6
6 3 x=
4 23
,02
Y-intercept - Let x=0
4 0-3y-6=0
-3y-6=0
-3y=6
6 y= 2
-3 0, 2
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Example
Find the x and y intercepts then graph using those points.
X-4y-8=0
x
y
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Summary
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Applications
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The graph gives the median age of the
US population in the indicated year. The
data is displayed as a scatterplot with two
points on the line indicated. Find the
equation of the line, in order to
make predictions of the US
population in the future.
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Now we will use the equation to predict
the median age of the US population in 2010.
That means we will substitute in 40 for the x.
The reason we use 40 is the initial date was 1970.
If we add 40 to 1970 we will get 2010.
y=0.265x+27.35
y=0.265(40)+27.35
y=37.95
This means that the median age of the US population
will be 37.95 in 2010.
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Example
Diameter 8 10 12 16
Price 6.40 8.00 9.60 12.80
The local pizza shop has a special sale on pizzas. Write
the slope-intercept equation of the line that describes the
price as a function of the diameter of the pizza.
If this company decides to make an 18 inch pizza, how
much should they charge?
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Graphing Calculator-Linear Regression
D $
8 6.40
10 8.00
12 9.60
16 12.8
Take the data from the previous pizza problem.
Put the data into List1 & List2 in the graphing
calculator.
To do that Press STAT,
then 1 for Edit. Type in
the numbers.
Press STAT, move the
cursor to the right to CALC,
then press 4 for LinReg.
The next screen gives you the values of
a and b for the equation.
The equation is y=.8x.More on the next slide.
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To see the scatterplot of the data, we need to
change the Window. Press WINDOW, and
type in what you see at left.
Press the GRAPH key. You will see the scatterplot
at left. The equation y=.8x will go through these
points. Press Y= and type in the equation. Press
GRAPH to see the line and scatterplot.
Graphing Calculator-Linear Regression continued
Press 2nd Y= to get STAT PLOT. First make
certain that all plots are off by pressing 4. Then
return to STAT PLOT and press Plot1. On the
word ON press ENTER.
Cursor down and press the appropriate
1
2
keys so
you get what you see in the picture at left. L is
obtained by pressing 2nd then 1. L - 2nd then 2.
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(a)
(b)
(c)
(d)
Find the equation of the line in slope-intercept form for a line that passes through (0,-4) and has a slope of -2.
2 4
4 2
2 4
2 4
y x
y x
y x
y x
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(a)
(b)
(c)
(d)
Find the equation of the line in slope-intercept form of the line that passes through (-3,-2) and (0,-2).
x 3
2
0
2 3
y
y
y x
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(a)
(b)
(c)
(d)
What is the slope of the line 3x - 7y – 4 = 0.
7
37
44
73
7